Q.Reduce the following Boolean Expression to its simplest form using K-Map: F(P,Q,R,S) = Σ(0,1,2,3,5,6,7,10,14,15)

The key idea is to use a Karnaugh Map (K-Map) to visually group adjacent minterms (1s) in powers of two, thereby simplifying the Boolean expression to its minimal Sum-of-Products (SOP) form.

1. Construct the K-Map: A 4-variable K-Map is drawn, and '1's are placed in cells corresponding to the given minterms: $0, 1, 2, 3, 5, 6, 7, 10, 14, 15$.

         RS
   PQ  00 01 11 10
   00 | 1  1  1  1  (0,1,3,2)
   01 | 0  1  1  1  (4,5,7,6)
   11 | 0  0  1  1  (12,13,15,14)
   10 | 0  0  0  1  (8,9,11,10)
   

2. Identify Essential Prime Implicants:

   • Quad (0,1,2,3): This group covers the entire first row. It simplifies to $P'Q'$. (Essential due to minterm 0).
   • Quad (1,3,5,7): This group covers cells $P'Q'S$ and $P'QS$. It simplifies to $P'S$. (Essential due to minterm 5). …

Concept and Intuition

Boolean algebra is fundamental to digital circuit design, but simplifying complex expressions can be tedious using algebraic theorems alone. Karnaugh Maps (K-Maps) provide a graphical method to simplify Boolean expressions, especially for up to 4 or 5 variables.

The core idea behind a K-Map is to arrange minterms (product terms where each variable appears exactly once, either in its true or complemented form) in a grid such that adjacent cells differ by only one variable. This arrangement is based on Gray code, which ensures that logical adjacency (differing by one variable) corresponds to physical adjacency on the map (next to each other, or wrapping around the edges).

When we group adjacent '1's (representing true minterms) in powers of two (2, 4, 8, etc.), we are essentially finding common terms that can be factored out, thereby eliminating redundant variables. For example, if we have $AB'C + ABC$, we can factor out $AC(B' + B) = AC(1) = AC$. On a K-Map, $AB'C$ and $ABC$ would be adjacent, and grouping them directly yields $AC$. The goal is to cover all '1's with the largest possible groups, using the fewest number of groups, to achieve the most simplified Sum of Products (SOP) expression.

Step-by-Step Solution

We are given the Boolean expression $F(P,Q,R,S) = \Sigma(0,1,2,3,5,6,7,10,14,15)$. This means the function outputs '1' for these minterms and '0' for all others.

1. Construct the K-Map

First, we draw a 4-variable K-Map. The variables $P$ and $Q$ define the rows, and $R$ and $S$ define the columns. The rows and columns are labeled using Gray code (00, 01, 11, 10) to ensure adjacency. Each cell in the K-Map corresponds to a unique minterm. We place a '1' in the cells corresponding to the given minterms and leave the other cells blank (implicitly '0').

The K-Map with minterm numbers and filled '1's:

2. Identify and Group Adjacent '1's

Now, we identify groups of adjacent '1's. The rules for grouping are:

• Groups must be rectangular or square.
• Groups must contain $2^n$ cells (1, 2, 4, 8, 16, etc.).
• Groups should be as large as possible.
• Every '1' must be covered by at least one group.
• Overlapping groups are allowed if they help form larger groups or cover '1's more efficiently.
• Groups can wrap around the edges of the map (top-to-bottom, left-to-right).

Let's identify the groups:

• Group 1 (Quad of 4): The entire first row (minterms 0, 1, 3, 2) forms a group of four '1's.

  • In this group, $P$ is always 0 ($P'$) and $Q$ is always 0 ($Q'$). $R$ and $S$ change their values, so they are eliminated.
  • This group simplifies to $P'Q'$.

• Group 2 (Quad of 4): The four '1's in the rightmost column (minterms 2, 6, 14, 10) form a vertical wrap-around group.

  • In this group, $R$ is always 1 ($R$) and $S$ is always 0 ($S'$). $P$ and $Q$ change their values (00, 01, 11, 10), so they are eliminated.
  • This group simplifies to $RS'$.

• Group 3 (Quad of 4): The four '1's in the second column (minterms 1, 3, 5, 7) form a vertical group.

  • In this group, $S$ is always 1 ($S$). $P$ is always 0 ($P'$). $Q$ and $R$ change their values, so they are eliminated.
  • This group simplifies to $P'S$. …